Conventional notation is limited mostly to conventional music. It is possible to notate rhythms as played, but next to impossible to read back the results.
On the other hand many of the Digital Audio Workstation programs (DAWs) do allow you to record and play Midi based music in non-western scales and even microtonally.
Many Synth modules allow a variety of alternate scales to Even Temperament, such as Pelog and Slendro, to be selected as well as Just intonation or Pythagorean.
Microtones can be reproduced using a controller lane to automate the pitch bend controller.
The resulting display will show accurate timing, including note start and end timing, and basic pitch plus modification as well as individual note velocity on controller lanes.
It will be in "Piano-Roll" format- which I can understand quite easily, though I have never used it to sight read from!
Cheers
Dave